10.4 Markovian methods

There is a family of related methods, collectively known as the Markov chain Monte Carlo or MCMC methods. These are stochastic methods that allow us to get samples from the true posterior distribution as long as we can compute the likelihood and the prior point-wise. You may remember that this is the same condition we needed for the grid method, but contrary to them, MCMC methods can efficiently sample from higher-probability regions in very high dimensions.

MCMC methods visit each region of the parameter space following their relative probabilities. If the probability of region A is twice that of region B, we will obtain twice as many samples from A as we will from B. Hence, even if we are not capable of computing the whole posterior analytically, we could use MCMC methods to take samples from it. In theory, MCMC will give us samples from the correct distribution – the catch is that this theoretical guarantee only holds asymptotically, that is, for an infinite...